Numerical multiscale methods are designed to overcome this issue by resolving the micro-structures only locally to solve small cell problems that yield the fine-scale information required to build an effective coarse-scale system, which then can be solved with modest computational effort.
Some of these multiscale methods, such as the Heterogeneous Multiscale Method (HMM) [1]},
are based on ideas from mathematical homogenization and aim at computing effective coefficients for an appropriate coarse-scale equation.
Other approaches, such as the Multiscale Finite Element Method (MsFEM) [2]}, [3]} or the Generalized Finite Element Method (GFEM) [4]}, instead construct coarse-scale elements that incorporate the local fine-scale features of the solution and then approximate the solution in the space spanned by these multiscale elements.
